#508419

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

554 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,39)(26,40)(27,38)(28,37)(29,34)(30,41)(31,32)(33,36)(35,42)(43,44);(6,9)(10,12)(13,15,16,25)(14,19,20,23)(17,30,28,39)(18,33,29,27)(21,24,37,41)(22,38,34,36)(26,43,42,32)(31,35,44,40);(2,4)(7,11)(10,12)(13,14)(15,24)(16,20)(17,31)(18,34)(19,30)(21,32)(22,29)(23,39)(25,41)(26,36)(27,35)(28,44)(33,40)(37,43)(38,42);(5,7,8,11)(6,10,9,12)(14,20)(19,23)(21,37)(22,34)(24,41)(31,44)(35,40)(36,38)

Polycyclic presentation:

2^9|a^2=efi,b^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415615);